Teacher resources and professional development across the curriculum
Teacher professional development and classroom resources across the curriculum
Defining Problem Solving
|Introduction | Introducing New Content Via Problems | A Positive Problem-Solving Disposition | Problem-Solving Techniques and Examples | Working Backwards | Technology | The Teachers' Role | Your Journal|
Teachers should introduce a variety of problem-solving techniques and encourage their use by students. Several techniques are outlined here with example problems. Collecting strategies and learning how to teach them well should be an ongoing activity for mathematics.
Generating and Organizing Data
The Staircase problem that started this session is a good example of generating data and then organizing it. In that case, a figure could be made and inspected. Results could be tabulated, and this in turn could lead to an understanding of the general case.
Other examples of generating and organizing data can be found in the Tile Patterns problems in Session 2 of this course.
Here is a problem that relies on generating possibilities and testing them:
Problem: Three Sisters
Three sisters were discussing their favorite positive integers. When their mother asked what numbers they had chosen, Lara said, "The product of our numbers is 36."
"So what are the numbers? I can't tell from that information," replied their mother.
"The sum of the numbers is my age," said Mara.
"I still can't tell," their mother said.
"My number is bigger than both of theirs," said Nara.
"Now I can tell!" said their mother.
Here is one problem-solving technique.
First, find the factors of 36 to generate possible solutions for this problem. We are looking for three integers with a product of 36:
The answers are as follows:
1, 1, 36
1, 2, 18
1, 3, 12
1, 4, 9
1, 6, 6
2, 2, 9
2, 3, 6
3, 3, 4
The mother said that the product of the three numbers was not enough information for her to discern the actual numbers, so the sisters gave additional information: The sum is Mara's age.
This means we need to know the sum of each of the triples:
1, 1, 36 sum = 38
1, 2, 18 sum = 21
1, 3, 12 sum = 16
1, 4, 9 sum = 14
1, 6, 6 sum = 13
2, 2, 9 sum = 13
2, 3, 6 sum = 11
3, 3, 4 sum = 10
The mother knows Mara's age but cannot find the numbers, even though she knows both their sum and product.
In order to solve this problem, you must really step back from the process and think about the given conditions. Under what conditions could you know the sum and product of three numbers and still not know what the numbers are?
There must be more than one set of numbers with a particular sum.
There is only one repeated sum.
Mara says that her number is bigger than both of her sisters' numbers.
The numbers are 2, 2, and 9.
This process of talking through the possibilities and eliminating the ones that do not work more or less "walks" the solver through the solution process. The ability to ask yourself questions and then give yourself the answers to these questions is a valuable problem-solving skill. When students work in groups, they can question each other.
Watch the video segment (duration 1:24) at left to hear reflections from teacher Carlos Cabana.
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